17 APPLICATIONS: II DUAL OF L1 201 17. APPLICATIONS: II. DUAL OF Ll (13.17.1 ) Let y. be a positive measure on X. For each function g e ^f^(X, u) (resp. g e ^^(X, /*)), the mappingdg:f\-* \fg dp, is a continuous linear form on the space JSf i(X, u) (resp. JS?c(X> u)) with respect to the topology defined by the norm Nx. The corresponding continuous linear form 9g: f\— > ti(fg) on the Banach space L^(X, u) (resp. L£(X, a)) has norm (5.7.1) equal to N00(^). Conversely, every continuous linear form on J2?i(X, u) (resp. cSfc(X, u)) is of the form f ^r^i fg du, where g is a function belonging to Jzf£P(X, u) (resp. ^^(X, //)), and the equivalence class of g is uniquely determined. We shall give the proof for JSf^ • The fact that 6g is a continuous linear form on J*?c» and the inequality \\Sg\\ ^ N^^), are consequences of (13.12.5). To prove that \\8g\\ = N «,(#), we may assume that g is not /^-negligible (in other words, that N00(^) > 0), since the assertion is trivial otherwise. Let a be any real number such that 0 < a < N00(^). Then there exists a compact subset K, of X, with measure /^(K) > 0, such that g \ K is con- tinuous and \g(x)\ > a for all x e K ((13.9.9) and (13.9.4)). For each a > 0 there exists a finite covering (U£) of K by open sets in K, such that the oscilla- tion of g in each Uf is ^8 (3.16.5). Hence there exists a partition of K into a finite number of integrable sets Ak such that the oscillation of g in each Afc is ^e (13.9.12.1). At least one of these sets, which we shall denote by A, has measure u(A) > 0. Let b be one of the values taken by g in A. We have \b\ > a and \g(x) — b\ :g e for all x e A. Now put clearly fe £?lc and Nx(/) = 1. Also we have f 5 f 5 — gcpA dp. = \b\u(A) + — (g - b)(pA dp. J PI J \b\ so that |/^rfjU ^.a — e. This proves our assertion, because a>0 and a < NOO(#) were arbitrary. Conversely, let u be a continuous linear form on 3?^. Then there exists a real number c> 0 such that \u(f)\ g c - N^/) for all/6 &lc (12.14.1). If K is any compact subset of X, it is known (13.11.6) that the topology induced on Jf C(X; K) by that of J$?c(x> A*) is coarser than the topology defined by the norm ||/||. Hence the linear form u is continuous on each of the Banachis another positive measure on X